The edge weight only depends on the spin carried by the edge and is simply the dimension of the corresponding representation up to a sign. The face weight is a mere parity factor. The bubble weight carries the non-trivial dynamical information of the model.
Such a structure with a cellular complex dressed with representations and intertwiner states and a probability amplitude defined as the product of local amplitude for each cell depending solely on the algebraic data it carries is called the local spinfoam ansatz for a path integral over discrete quantum geometries. It has been shown in [1]}, [2]} that they are Feynman diagram amplitudes of non-local non-commutative field theories, referred to as group field theories or tensorial group field theories (see [3]} on recent studies of those Feynman diagrams and the renormalization of such field theories).
